Shape parameters measurement of ultralight mirrors

Optik 121 (2010) 1881–1884

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Shape parameters measurement of ultralight mirrors Miroslav Pech , Duˇsan Manda t 1, Miroslav Hrabovsky´ 2, Miroslav Palatka 3, Petr Schova nek 4 Joint Laboratory of Optics of Palacky University and Institute of Physics of Academy of Sciences of the Czech Republic, Tˇr. 17. listopadu cˇ. 50a, Olomouc, Czech Republic

a r t i c l e in fo

abstract

Article history: Received 23 January 2009 Accepted 9 May 2009

Two measuring techniques are presented in this paper. These techniques are able to measure and process the shape of the reflecting surfaces. In the first part of the paper the theory and modifications of the optical, mechanical and software design for the Hartmann wavefront analyzer are described. This new method is applied to the non-contact measurement of concave-mirrors shape and objectively defines the shape and specifies the difference of the segment shape from an ideal surface with micrometer accuracy. This difference determines essentially the mirrors image quality. In the second part of the paper attention is concentrated on the other main factor that affects image quality— surface roughness. Thereinafter, physical background and usage of the measuring system and results are presented. & 2009 Published by Elsevier GmbH.

Keywords: Hartmann test Roughness Scattering BRDF Mirror shape

1. Introduction During manufacturing of the reflective elements it is necessary to control two main parameters, that influence efficiency and image quality of the elements. These are the ‘‘micro and macro’’ shape of the element. Therefore the quantitative research of the roughness and element surface shape is required. Our institute (Joint Laboratory of Optics) produces segmented mirrors for the Pierre Auger Observatory fluorescent detector [1]. We followed the methods described below as the quantitative quality measurement of the spherical mirror segments.

the Pierre Auger Observatory mirror segment shape. These segments are unique because they are very light and ultrathin. The production of mirrors in our laboratory is based on standard technological operations used commonly in the optical industry (cutting, drilling, milling, grinding and polishing) with regard to the production of extremely thin elements to minimize weight. This fact can cause segment shape instability in the production process. The method objectively defines the shape and specifies the difference of the segment shape from an ideal surface with micrometer accuracy.

2.1. Designed experimental setup 2. Macroscopic shape measurement The theory, optical, mechanical design modification of the classical Hartmann wavefront analyzer and software design are presented here. This new method is applied to the non-contact method of concave-mirror shape measurements. The Hartmann test [2] was invented in the last century to perform optical metrology. Subsequently these sensors have been adapted to a wide variety of applications including adaptive optics, ophthalmology, and laser wavefront characterization. We use the principle of this test as root of our designed method for measuring  Corresponding author. Tel.: +420 585631520; fax: +420 585631531.

E-mail addresses: [email protected] (M. Pech), [email protected] ´ [email protected] (D. Manda t), [email protected] (M. Hrabovsky), (M. Palatka), [email protected] (P. Schova nek). 1 Tel.: +420 585631573, fax: +420 585631531. 2 Tel.: +420 585631501, fax: +420 585631531. 3 Tel.: +420 585631516, fax: +420 585631531. 4 Tel.: +420 585631503, fax: +420 585631531. 0030-4026/$ - see front matter & 2009 Published by Elsevier GmbH. doi:10.1016/j.ijleo.2009.05.008

In the measuring design we use a simple axis optical setup (Fig. 1). The setup consists of a laser diode as a high-intensity point source. The light beam is very intensive and very homogeneous. Disturbing speckle pattern is possible to subtract from the final image. After the mirror and pellicle beamsplitter reflections the beam crosses the transparent spatial modulator, which is located between two cross polarizers. The combination of the two cross polarizers and phase modulator forms an amplitude modulator.

2.2. Data processing There are five steps of analysis process: digital pattern processing, determination of the pinhole images positions, conversion to wavefront slopes, and wavefront reconstruction. Spatial modulator produces step-by-step binary masks and wavefront slopes are measured in several points in each step.

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Fig. 3. Schematic layout of the measurement geometry, where S is a point light source, M is incident point on the mirror surface and H is center of the mask pinhole. The dashed line is a normal of the mirror surface.

and Fig. 1. Layout of the designed measuring setup, the dashed line is a traced beam that goes through the center of the mask pinhole.

P

yc;k ¼

yi;j Ii;j

i;j2AoIk

P

Ii;j

ð2Þ

i;j2AoIk

where k is a spot number and the summation is taken over the pixels assigned to the spot k in the Area-of-Interest AoIk. Then the given spot centers are sorted and assigned to the pinholes on the mask.

Fig. 2. Process of the mirror surface scanning. (a) Pattern on the spatial modulator, (b) CCD snapshot, ‘‘pinhole images’’, (c) picture after background subtraction and filtering, (d) final spot centers, and (e) all spots in one picture.

2.2.3. Conversion to mirror shape slopes There is an easy way to find the wavefront gradient or mirrorshape normal orientation. If the point source position, the incident reflection point position and the position of the given pinhole are given, then it is just an easy geometric problem to find the normal of the mirror surface in the incident point (Fig. 3). In this way it is possible to find Eq. (2), the surface gradient of each incident points: ~ n¼

Fig. 2 illustrates the example of a surface tested in 30 radial symmetry patterns with 32 points. The spatial modulator creates ‘‘pinhole images’’ (see Fig. 2), which are detected by the CCD camera. In our setup we use patterns with dynamically changing radially distributed pinholes. These patterns are advantageous for its central symmetry and easy processing in a polar coordinate system. This designed mask is able to detect the most common flaws of the mirror surface, zonal errors, and concentric hills and valleys. In further sections measuring sequences are described. The whole measuring process is divided into five steps, which will be described in the following sections. 2.2.1. Digital pattern processing The CCD camera creates an image of pinholes. This image consists of diffracted spots and background (the mask is a little translucent outside the pinholes and the CCD chip produces some noise). After morphological operations, noise is filtrated and small noise areas are removed subsequently. The final result is a pattern with the intensity profile of the pinholes images with intermediate space value equaled to zero. 2.2.2. Spot positions The pinhole image positions are computed from the modified pattern. The location of the focal spots is determined from the light distribution on the detector array. The spot positions xc,k and yc,k are commonly determined by the first moments for a sampled irradiance distribution with measured pixel intensities Iij: P xi;j Ii;j xc;k ¼

i;j2AoIk

P

Ii;j

i;j2AoIk

ð1Þ

~ ~ SM þ HM : ~ j~ SM þ HMj

ð3Þ

2.2.4. Wavefront reconstruction Once the local wavefront slopes have been determined, the wavefront can be reconstructed by performing a type of integration on gradient measurements. Discussion is limited to considering one radial part of the image, but the rest can be treated in the same way. The measured two-dimensional shape is designated by the variable F(x, y). The two primary types of wavefront reconstruction algorithms are zonal and modal. The zonal wavefront reconstruction is a type of numerical integration. The modal wavefront reconstruction fits the data to a set of orthogonal surface polynomials, e.g. Zernike polynomials in the Modal Zernike Reconstructor [4]. In this work our attention is focused on the zonal reconstruction, specifically on linear integration. 2.2.5. Linear integration The algorithm of the simplest surface reconstructors (it is possible to use more complicated zonal reconstructors, e.g. Southwell [3]) begins at the center of the pattern and defines the mirror shape height at this integration area as zero. The height profile of the mirror surface Fn in the next adjacent integration area along the scan direction r is calculated as the previous wavefront height Fn1 plus the average slope of the current and the previous integration area times the aperture separation s:

Fn ¼ Fn1 þ

  @Fn1 @Fn s þ : @r @r 2

ð4Þ

2.3. Achieved results Fig. 4 shows our achieved results of the shape measurement. The measured sample was the hexagonal mirror segment used in the fluorescent detector. The radius of the circumscribed circle is 620 mm and thickness is 12 mm. It is evident from this figure that

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frequencies fi, which are related to the scatter angle ys by the onedimensional grating equation sinðys Þ  sinðyi Þ ¼ fi l;

ð5Þ

where yi is incident angle and l is wavelength of the incident light. The BSDF function is the relation between the light power received within a solid angle in a certain scattering direction depending on the beam incidence angle and its power. This determines the position and the magnitude of the diffraction orders Fig. 4. Measurement results of the hexagonal mirror segment shape—difference between ideal spherical and real shape.

the mirror has an astigmatic aberration. This deformation is caused by non-optimal sticking of the ultrathin segment to the polishing base. The mirror-segment shape is deformed this way after unstick from the base. This designed method is quantitative research of the concave mirror shape. It is a straightforward technique for measuring of large, even highly aberrated, concave mirrors with micrometer accuracy. The usage of the amplitude modulator brings flexibility to the measurement method: speed versus resolution optimization. It is possible to scan the whole mirror surface or only the problematic part of the surface.

3. Microscopic shape measurement Because of diffraction phenomenon at the entrance of the optical element, the power of the input beam is spread onto a small area instead of a single point. The diffracted angular deflection with respect to purely specular directions is very small. This enlargement depends on the beam diameter D (it is the telescope aperture diameter) and the wavelength l, so most of the power falls inside a solid-angle circle of an angular radius 1.22l/D. This phenomena is known as the Airy disk. This is the minimum image dimension for perfect conditions and it is used as a natural scale in our discussion. If the surface is not perfect and the defects are smaller than a wavelength, some additional power will fall outside the Airy disk. This type of scattering yields to mathematical analysis, which allows computing the relation between the physical shape of the defects (microroughness) and the resulting optical scattering. This is the typical case of a good optical surface such as a clean optical mirror.

BSDFðys Þ ¼

The scattering properties of the surface are measured by a scatterometer. This instrument throws essentially collimated beam on the test surface at a defined incidence angle, and one or several detectors detect and capture the scattered light. These detectors are mounted on arm of the goniometer, where the scattering angle can be varied. The scattering solid angle is defined by baffles and stops. One of the methods on how to determine the angle-resolved scattering is known as Bidirectional Scattering Distribution Function (BSDF). Relatively smooth surfaces (peak to valley ˚ will reflect most of the light into the distance is less than 500 A) zero order (the specular direction) and diffract small fractions of the light to the +1 and 1 orders. Light diffracted into the second order can be neglected for gratings with the smooth surfaces. In this case we can describe the surface by sum of gratings with

ð6Þ

where ys is the scattering direction (from the normal), Pi the power in the incident beam, dP the power in the scattered beam inside the solid angle dOs around the direction ys. Therefore, BSDF is physically nothing more than the redistributed energy scattered into a given solid angle. The magnitude of the first-order light is determined by the sinusoidal amplitude and frequency, while the position (angle of diffraction) is determined by the grating frequency and direction. Any arbitrary surface composed of many sinusoidal surfaces should then diffract in many directions where each direction and magnitude define a sinusoidal component present on the surface. Measurements of the magnitude and direction of the scattered light can be used to calculate the amplitude and frequency of the sinusoidal components present once it is known exactly how sinusoidal gratings diffract their light. Measurement of these quantities amounts to measurement of the surface PSD. The PSD function then is calculated from the BSDF [5] and is a measure of the scattered power per unit of spatial frequency in units of A˚ 2 mm2: 4

PSDðf Þ ¼

108 l BSDF : 16p2 cos yi cos ys Q

ð7Þ

For s-polarization, the factor Q is approximated by the specular reflectance of the surface. This function defines the surface statistics but not the exact surface profile. However knowing the PSD of a given surface allows determination of RMS roughness in a straightforward manner. The RMS roughness Rq between spatial frequencies fmin and fmax from a one-dimensional power spectral density function PSDISO is given in Eq. (8). If the surface is isotropic, PSD(f) may be integrated around the azimuthal angle to obtain an isotropic PSDISO function sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z fmax

Rq ¼ 3.1. Measuring of the surface scattering properties

dPðys Þ ; Pi dOs cosðys Þ

2

PSDISO ðf Þ df :

ð8Þ

fmin

3.2. Instrumentation Instruments designed to obtain BSDF measurements must have either a single moving receiver or multiple receivers arrayed around the sample at scatter angles necessary for the desired measurement. The receiver consists of apertures set and a scatterlight detector. Sometimes a lens, field stop, bandpass filter and polarizing elements are added. Our measuring system SMS’s Complete Angle Scan Instrument (CASI) has both moving detector and multiple-fixed detector instruments that measure scatter in the plane of incidence (defined by the incident beam and the sample normal). The receiver moves around the sample (3601 around) as is shown in Fig. 5. The single receiver in the CASI instrument is able to move with very small steps (0.0071); in this way the measurement can be very close to the specular reflection.

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Fig. 5. Outline of the CASI system.

3.3. Achieved results In this section our attention will be focused on scatter measuring, concretely microshape parameters of the mirror segments. This way it is possible to reach a parameter that quantitatively describes roughness characteristic of the reflective surface. Fig. 6a shows the BSDF Eq. (6) plot of two randomly polished mirrors. As explained above, this is a plot of scattered intensity versus angle. Considering Fig. 6 we can easily see that the BSDF function of mirror A is higher than the BSDF function of mirror B in higher angular parts. It indicates that mirror A scatters more light than mirror B. Without any mathematics, we can immediately note that the surface of mirror B is smoother than the mirror A surface. We can support this assumption with PSD calculation of Eq. (7). Fig. 6b illustrates mirrors A and B PSD functions (Eq. (7)). Mirrors A and B have approximately the same scatter near the specular angle (for optical surfaces, higher scatter near specular results in a degradation of the resolution of an object). Mirror A produces more scatter light than mirror B at higher angles (higher scatter light at higher angles often produces glare or ‘‘noise’’ in an optical system). We can, therefore, deduce that mirror A surface has a greater number of short surface characteristics than mirror B. Numerically we can describe this result with lower number in exponent in functional fit of the PSD functions (Fig. 6b). Using Eq. (8) we can characterize reflective surface roughness of mirror A and mirror B with one quantity RMS surface roughness. For these surfaces we can obtain the following values: RMS(mirrorA) ¼ 45.3 ˚ We use these values as a criterion of A˚ and RMS(mirrorA) ¼ 35.1 A. quality of our mirror surface.

4. Conclusion The paper describes a straightforward technique for quality measuring of large, optical imaging as well as highly aberrated concave mirrors used for energy collection. The designed method can make rapid measurements using the Hartmann mirror shape sensor. This method can be used as fast quality test in optics shops with micrometer accuracy. The measurement of the microshape quality was also presented. It was shown that we can obtain not ˚ ¨ only the value of the RMS roughness in Angstr oms accuracy but also information about the surface spatial frequencies. These results help us to improve our production technology.

Fig. 6. (a) BSDF function of the two reflective samples compared with device signature (BSDF values measured without sample) and (b) PSD function of the same samples.

Acknowledgement These results of the project 1M06002 were supported by The Ministry of Education of the Czech Republic and results of the project KAN301370701 were supported by Project of Academy of Science of the Czech Republic. References [1] The PAO collaboration, The Pierre Auger Observatory technical design report, /http://www.auger.orgS, 2001. [2] D. Malacara, Optical Shop Testing, Wiley, New York, 1992. [3] W.H. Southwell, Wave-front estimation from wave-front slope measurements, J. Opt. Soc. Am. 70 (1980) 998–1006. [4] A. Gavrielides, Vector polynomials orthogonal to the gradient of Zernike polynomials, Opt. Lett. 7 (1982) 526–532. [5] J.C. Stover, Optical Scattering Measurement and Analysis, McGraw-Hill, New York, 1990.